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Nuclear Pairing Gaps and Neutron Star Cooling universe Article Nuclear Pairing Gaps and Neutron Star Cooling Jin-Biao Wei ∗ , Fiorella Burgio and Hans-Josef Schulze INFN Sezione di Catania, Dipartimento di Fisica e Astronomia, Università di Catania, Via Santa Sofia 64, 95123 Catania, Italy; [email protected] (F.B.); [email protected] (H.-J.S.) * Correspondence: [email protected] Received: 13 July 2020; Accepted: 5 August 2020; Published: 8 August 2020 Abstract: We study the cooling of isolated neutron stars with particular regard to the importance of nuclear pairing gaps. A microscopic nuclear equation of state derived in the Brueckner-Hartree-Fock approach is used together with compatible neutron and proton pairing gaps. We then study the effect of modifying the gaps on the final deduced neutron star mass distributions. We find that a consistent description of all current cooling data can be achieved and a reasonable neutron star mass distribution can be predicted employing the (slightly reduced by about 40%) proton 1S0 Bardeen-Cooper-Schrieffer (BCS) gaps and no neutron 3P2 pairing. Keywords: neutron star; nuclear superfluidity; nuclear equation of state 1. Introduction A very important effect of nuclear superfluidity in a neutron star (NS) is the suppression of standard neutrino cooling processes and the appearance of new ones, which then compete with each other [1–4]. The superfluidity has therefore a decisive influence on the temperature evolution of an isolated NS, and this can be compared with known observational data in order to deduce constraints on the various pairing gaps and also on the nuclear equation of state (EOS). This article is dedicated to this problem, and we will perform a detailed analysis of NS cooling with the above goal. It is currently still not clear whether all NSs have a purely nucleonic substructure, that is, can be considered to be built of individual neutrons and protons, or whether heavy NSs hide exotic baryons like hyperons or even deconfined quark matter in their extremely dense interior. In this work we follow the simple first assumption and consider purely nucleonic NSs. We model their internal structure by a theoretical EOS that has been derived within the Brueckner-Hartree-Fock (BHF) many-body method, fulfilling all current constraints imposed by observational data from nuclear structure, heavy-ion collisions, NS global properties, and recently NS merger events [5–7]. Within this framework we then investigate the cooling evolution of NSs, and in particular the effect of the proton 1S0 and the neutron 3P2 pairing gaps. This investigation has been carried out in several previous publications [8–11], and we refine here our analysis regarding the constraints on the pairing gaps deduced from comparison with cooling data. In fact we have previously concluded that a good reproduction of all current cooling data is possible by assuming the Bardeen-Cooper-Schrieffer (BCS) gap in the proton 1S0 channel, but not allowing pairing in the neutron 3P2 channel. However, the BCS approximation disregards any medium effects on the gaps, which are not supposed to be small in the dense NS matter. An accurate quantitative theoretical computation of such effects is however still very difficult or impossible, and therefore we investigate in this work empirically the effects of such modifications on the cooling evolution in order to identify possible constraints that may be obtained in this way. In particular, we concentrate in this article on the NS mass distribution that can be obtained by comparing theoretical cooling curves with the currently available set of cooling data [12] (We do not yet utilize the very recent update of these data [13] in this work), and which is therefore a functional of the pairing gaps. Universe 2020, 6, 115; doi:10.3390/universe6080115 www.mdpi.com/journal/universe Universe 2020, 6, 115 2 of 16 This paper is organized as follows. In Section2 we give a brief overview of the theoretical framework, namely the BHF formalism adopted for the nuclear EOS, the various cooling processes, and the related nucleonic pairing gaps. Section3 is devoted to the presentation and discussion of the results for stellar structure, the cooling diagrams, and the derived mass distribution. Conclusions are drawn in Section4. 2. Formalism 2.1. Nuclear Equation of State The nuclear EOS of the model is derived in the framework of the Brueckner-Bethe-Goldstone theory, which is based on a linked-cluster expansion of the energy per nucleon of nuclear matter [14–16]. The basic ingredient in this many-body approach is the in-medium Brueckner reaction matrix G, which is the solution of the Bethe-Goldstone equation (¯h = c = 1) j12i Q h12j G(E; r, x) = V + V G(E; r, x) , (1) ∑ − − 1,2 E e1 e2 where V is the bare nucleon-nucleon (NN) interaction, E is the starting energy, and the multi-indices 1, 2 denote in general momentum, isospin, and spin. x = rp/r is the proton fraction, and rp and r are the proton and the total baryon density, respectively. The propagation of intermediate baryon pairs is determined by the Pauli operator Q and the single-particle (s.p.) energy 2 k1 e1 = e(1; r, x) = + U1 . (2) 2m1 The BHF approximation for the s.p. potential U using the continuous choice is U1(r, x) = Re ∑ h12jG(e1 + e2; r, x)j12ia , (3) (2) 2<kF where the matrix element is antisymmetrized. Due to the occurrence of U1 in Equation (2), the coupled system of Equations (1)–(3) must be solved in a self-consistent manner for several Fermi momenta of the particles involved. The corresponding BHF energy density is k2 1 e = ∑ 2 ∑ + Ui(k) . (4) i=n,p (i) 2mi 2 k<kF It has been shown that the energy and the nuclear EOS can be calculated with good accuracy in the Brueckner two-hole-line approximation with the continuous choice for the s.p. potential, since the results in this scheme are quite close to the calculations which include also the three-hole-line contribution [17–20]. In this scheme, the only input quantity needed is the bare NN interaction V in the Bethe-Goldstone Equation (1). In the present work, we use the Argonne V18 potential [21] as the two-nucleon interaction, supplemented by a consistent meson-exchange three-body force (TBF), which allows to reproduce correctly the nuclear-matter saturation point [22–25] and other properties of nuclear matter around saturation [26]. Further important ingredients in the cooling simulations are the neutron and proton effective masses, which we actually used in our previous simulations presented in Reference [8]. In the BHF approach, the effective masses can be expressed self-consistently in terms of the s.p. energy e(k) [27], m∗(k) k de(k) −1 = . (5) m m dk Universe 2020, 6, 115 3 of 16 As found in [8], their effect can be absorbed into a rescaling of the pairing gaps that we also employ in this paper, and therefore we simply use the bare nucleon mass here. This is also convenient for comparison with other works that use bare masses. For completeness, we mention that the BHF method provides the EOS for homogeneous nuclear −3 matter, r > rt ≈ 0.08 fm . For the low-density inhomogeneous crustal part we adopt the well-known −3 Negele-Vautherin EOS [28] for the inner crust in the medium-density regime (0.001 fm < r < rt), and the ones by Baym-Pethick-Sutherland [29] and Feynman-Metropolis-Teller [30] for the outer crust −3 (r < 0.001 fm ). The transition density rt is adjusted to provide a smooth transition of pressure and energy density between both branches of the betastable EOS [31]. The NS mass domain that we are interested in, is hardly affected by the structure of this low-density transition region and the crustal EOS: The choice of the crust model can influence the predictions of radius and related deformability to a small extent, of the order of 1% for the value of a 1.4-solar-mass NS, R1.4 [31–33], which is negligible for our purpose. Even neglecting the crust completely, NS radius and deformability do not change dramatically [34]. In order to illustrate the bulk properties of the V18 EOS thus obtained, Figure1 shows the resulting NS mass-radius and mass-central density relations obtained in the standard way by solving the TOV equations for betastable and charge-neutral matter. We remark that the value of the maximum mass Mmax = 2.34 M of the V18 EOS is larger than the current observational lower limits [35–38]. Regarding the radius, we found in Reference [39,40] that for the V18 EOS R1.4 = 12.33 km, which fulfils the constraints derived from the tidal deformability in the GW170817 merger event, R1.36 = 11.9 ± 1.4 km [6], see also similar compatible constraints on masses and radii derived in References [41–47]. The V18 EOS is also compatible with estimates of the mass and +0.15 radius of the isolated pulsar PSR J0030+0451 observed recently by NICER, M = 1.44−0.14 M and +1.24 +0.15 +1.14 R = 13.02−1.06 km [48], or M = 1.36−0.16 M and R = 12.71−1.19 km [49]. The figure also shows the density of the onset of direct Urca (DU) cooling, rDU, and the one of the vanishing of the p1S0 BCS gap, r1S0, to be introduced and discussed in the following, along with the pairing parameter sx. 2.5 s =1.4 x 1.2 2.0 1.0 ( , M ) 1S0 1S0 0.8 J0030+0451 ] 0.6 1.5 GW170817 0.4 M [M ( , M ) 1.0 DU DU DU 0.5 0.0 0.0 0.3 0.6 0.9 10 11 12 13 14 -3 [fm ] R [km] c Figure 1.
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